Geoscience Reference
In-Depth Information
Table 14 .17
Stepwise condi-
tional transformations for Red
Dog
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3. Calculate and model the directional variograms for each
of the transformed variables within each rock type.
4. Independently simulate transformed variables via sequen-
tial Gaussian simulation (Isaaks
1990
).
5. Back transform the simulated values (back-SCT) to return
to original units.
6. Validation of simulation results to confirm data, histo-
gram and variogram reproduction.
Once all variables within all domains were modeled, the
block models were merged to form multiple realizations
of the study area for uncertainty assessment and post-pro-
cessing. The methodology described differs only from the
common geostatistical Gaussian simulation in the use of the
stepwise conditional transformation (SCT), in place of the
conventional normal score transform. SCT is a multivari-
ate Gaussian transformation approach whereby the prima-
ry variable is transformed to a standard normal; all subse-
quent variables are successively conditioned to the previous
variable(s) based on probability binning.
The transform applies to collocated multivariate data
making them independent prior to simulation. Cross vario-
grams between transformed variables should be checked to
verify that spatial correlations are approximately zero. After
such verification, independent Gaussian simulation can pro-
ceed. Back transformation restores the complex relation-
ships between the multivariate data.
The need to consider seven variables simultaneously for
any one rock type poses a practical problem. The multivari-
ate stepwise conditional transform would require 10
7
com-
posites in order to have a minimum of 10 data per probability
class. To overcome this problem, a
nested
application of the
stepwise conditional transformation was implemented. Infer-
ence of a trivariate distribution would require approximately
10
3
or 1,000 data to define the conditional distributions with
a minimum of 10 data. This is more reasonable given the
number of composites available.
The transformation ordering for the stepwise conditional
transform will affect the reproduction of the variogram of the
simulated values. Thus, the most important variable or the
most continuous variable should be chosen as the primary
variable (Leuangthong and Deutsch
2003
), which, for Red
Dog, is Zn. To account for the other six variables, sets of
transformations were proposed (see Table
14.17
).
This case study is limited to eight geological domains
corresponding to four different ore type units in the Upper
and Median plates. These were chosen because they corre-
spond to a volume that includes both recently mined material
and material that will be mined in the near future.
The existing grade models were independently kriged
onto 25 × 25 × 25 ft blocks. The geostatistical models were
simulated at 12.5 × 12.5 × 12.5 ft resolution, and later up-
scaled to 25 ft blocks. The simulation is on a point-scale sup-
port. Thus, simulating at a finer resolution and then averag-
ing to larger blocks shows the variability of the block grades
more accurately; change of support is addressed numerically.
The six modeled benches span a volume that is 4,500 ft
wide (Easting) by 4,500 ft long (Northing) by 150 ft in the
vertical direction. The model consists of a total of 1,555,200
grid points. The simulations were be constructed on a by
rock type basis, and then merged. All comparisons shown
are global, corresponding to all domains combined.
Two types of data were used: composited drillhole data
and blasthole data. The simulations were done using 12.5 ft
composites. The geology model, originally coded on the
25 ft blocks, was recoded into the 12.5 ft model.
There were a total of 9,847 12.5 ft composites available
for the eight domains of interest. The term drillhole (DH)
refers to the 12.5 ft composites. DH data are at a nominal
100 × 100 ft spacing. For these same domains, there were
58,566 blasthole (BH) data available for model validation.
BH data are spaced data 10 × 12 ft spacing, and represent the
entire 25 ft bench.
14.4.2
Multivariate Simulation Approach
Conditional simulations were performed for seven variables:
Zn, Pb, Fe, Ba, sPb, Ag, and TOC. These seven variables
were modeled for each rock type using Gaussian simulation
with stepwise conditionally transformed variables. The main
steps of the simulation are:
1. Data declustering to obtain representative distributions
for each variable.
2. Transform data with SCT (Leuangthong
2003
; Leuangth-
ong and Deutsch
2003
; Luster
1985
; Rosenblatt
1952
) to
obtain independent Gaussian variables.
